Derivative

Using the definition,

Therefore,

For a positive integer $n$,

Hence,

Therefore,

Add and subtract $f(x+h)g(x)$:

Thus,

For

let

Then

so

Differentiate

Differentiate

Differentiate

Differentiate

Differentiate

Find the equation of the tangent line to

at $x=2$.

Find the local extrema of

If

the tangent line at $x=a$ is horizontal.

But this does not necessarily mean that $f(a)$ is a local maximum or minimum.

For example,

We have

but $x=0$ is neither a local maximum nor a local minimum.

A function can have a local maximum or minimum even when

does not exist.

For example,

has a minimum at $x=0$, but $f'(0)$ does not exist.

The derivative of $f'$ is called the second derivative:

It describes how the slope itself is changing.

If

the graph is concave up.

If

the graph is concave down.

If

then

while

If

the test is inconclusive.

An equation does not have to be written as $y=f(x)$.

For example,

Differentiating both sides with respect to $x$,

so

The quotient

is an average rate of change.

Taking

gives the instantaneous rate of change.

If $f$ is differentiable at $x=a$, then $f$ is continuous at $x=a$.

However, the converse is false.

For example,

is continuous at 0 but not differentiable there.

For finding extrema, do not consider only

A critical point can also occur where

does not exist, provided $f(x)$ is defined there.

So the better statement is:

On an interval,

and

For extrema, the change of sign is important:

This is false.

has

but no extremum at 0.

When finding an absolute maximum or minimum on

check:

not just the solutions of $f'(x)=0$.